Computerm Math. Appli¢. Vol. 24, No. 3, pp. 13-28, 1992 0097-4943/92 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyright~ 1992 Pergamon Pre~ Ltd ICOSAHEDRAL SYMMETRY AND THE QUINTIC EQUATION R. B. KIN(] Department of Chemistry, Uuivendty of Georgia Athe~m, Georgia 30602, U.S.A. E. R. CANFIELD Department of Computer Science, University of Georgia Athens, Georgia 30602, U.S.A. (Received J~lll 1990) A.bstract--An algorithm has been implemented on a microcomputer for solving the general quintic equation. This algorithm is bam~cl on the isomorphism of the As alternating Galois group of the gcmeral quintic equation to the symmetry group of the Jcosahedron, coupled with the ability to partition an object of ieosahedral symmetry into five equivalent objects of tetrahedral or octabedral symmetry. A detailed discussion is presented of the portions of this algorithm relating to these symmetry properties of the icosahedron. In addition, the entire algorithm is summarized. I. INTRODUCTION One of the classicalproblems of algebra is the solution of polynomial equations using an algorithm which calculates the roots of the equation from its coefficients.Such a formula for the roots of a quadratic equation is well-known to even beginning algebra students and contains a single square root. An analogous but considerably more complicated algorithm containing both square and cube roots is known for solution of the cubic equation [1,2]. Appropriate combination of the algorithms for the quadratic and cubic equations leads to an algorithm for solution of the quartic equation [1,2]. However, persistentefforts by numerous mathematicians before the early 1800's to find an algortithm for solution of the general quintic equation using only radicals and elementary arithmetic operations were uniformly unsuccessful. The reason for the futilityof these efforts was uncovered by Galois [3-5], who proved the impossibilityof solving a general quintic equation using only radicals and arithmetic operations. Galois' proof led to the birth of group theory and the concept of the symmetry of an algebraic equation relating to permutations of its roots [3-5]. The demonstrated impossibility of solving quintic equations using only radicals and simple arithmetic led the 19th century mathematicians following Galois to develop the mathematics of the more complicated functions required for an algebraic solution of the quintic equation. The necessary functions turn out to be Jacobi and/or Weierstrass ellipticfunctions [6] and the theta series used for their evaluation. The required relationships between such ellipticfunctions and the roots of a class of quintic equations derivable from the general quintic equation by Tschirn- haus transformations [7] were first established by Hermite [8,9], and developed subsequently by Gordan [10] into an algorithm for solution of the quintic equation. This algorithm was both simplified and clarified by Kiepert [11]. The nature of these algebraic efforts was clarified further by publication, in 1884, of Klein's classical book linking the symmetry of the icosahedron with algorithms for solution of the quintic equation [12]. This book attracted the attention of one of us (R.B.K), who for many years has been fascinated by the special features of icosahedral symmetry [13] in chemical contexts, such as chirality [14], boron hydrides [15] and quasicrystals [16]. These 19th-century algorithms for the solution of the general quintic equation were so com- plicated that there was no hope of implementing them in that precomputer era. Numerical, We are indebted to the U.S. Office of Naval Research for partial support of this work. Typeset by A,~4s-'lkX 13 14 R.B. KINa, E.R. CANFIELD rather than algebraic methods based on elliptic functions, evolved to satisfy practical needs for determining roots of quintic and higher degree equations [17]. However, such methods ignore the interesting symmetry aspects of such equations first discovered by Galois [3-5]. The concise presentation of an apparent algorithm for solution of a general quintic equation at the end of the 1878 paper by Kiepert [11] stimulated us to try this algorithm on a modern microcomputer. However, our initial attempts soon revealed a number of ambiguities and even errors of the ex- pected type for an algorithm which had never been actually implemented because of the lack of the necessary computer technology at the time of its development. We therefore soon real- ized that, contrary to the impression given by Cole [18] as long ago as 1886, a really complete workable algorithm for solution of the general quintic equation by elliptic functions, rather than numerical methods, was not contained in any of the 19th century or more recent literature. We therefore undertook a deeper study of the underlying rather complicated mathematics. Useful surveys of much but not all of the necessary algebra for solution of the general quintic equation are presented in 20th century, but now rather old, algebra texts by Dickson [19] and Perron [20]. However, even these less ancient writeups lack much of the necessary information for a complete algorithm for solution of the general quintic equation. We have now filled in all of the gaps and have a complete computer-tested algorithm for solution of the general quintic equation by algebraic rather than numerical methods. The underlying mathematics of this successful algo- rithm is long and complicated. This paper focusses on the aspects of the algorithm based on the icosahedral symmetry of the general quintic equation. In addition, a summary of the complete algorithm is presented, so that it can be implemented by an interested reader on an appropriate microcomputer without referring to the literature. Future publications are planned to discuss other aspects of this algorithm. 2. THE SYMMETRY OF ALGEBRAIC EQUATIONS Consider an algebraic equation of degree n f(x) = a0x n + al x "-t +"'+ an = Za"-iz i = 0, (2.1) i----1 where ak (0 < k <: n) is rational. Without loss of generality, a0 can be taken to be unity since ak (1 <__ k _< n) can be adjusted appropriately. The polynomial f(z) is said to be factorable if it can be expressed as a product of polynomials of lower degrees with rational coefficients. The Galois group, G, of an algebraic equation is a permutation group of its roots [3-5]. If the polynomial of the equation is non-factorable, then the corresponding Galois group, G, is transitive, i.e., for any pair of roots zi and zk, there is an operation g in G such that zi is permuted to xk. The structure of the Gaiois group, G, of an algebraic equation relates to the types of functions that are needed to determine its roots. Consider a group G and a subgroup H, with g E G and h E H. If g is fixed and h is varied over the elements of H, then all elements of the form g hg -1 form another subgroup of G which is said to be conjugate to H. A normal subgroup N with n E N is one which is self-conjugate, i.e., g ng -1 generates N, and we write N~G. Subgroups H induce an equivalence relation on the group G partitioning it into various equivalence classes, two elements g and g~ of G being regarded as equivalent when g (g~)-t E H. A normal subgroup N as defined above consists of entire conjugacy classes of G. The conjugacy class headings of the character table of G are thus useful for finding the normal subgroups of G. If N is a normal subgroup of G, the quotient group G/N is well-defined and relates to the type of functions needed to determine the roots of an equation with the Galois group G. A group G with only the identity group C1 as a normal subgroup is called a simple group. Simple permutation groups tend to be groups of high order relative to the number of objects being permuted. For example, the alternating groups An, consisting of all n!/2 even permutations of n objects, are simple groups for n >_ 5. The smallest non-trivial simple permutation group is thus the alternating group As, which is isomorphic with the icosahedral pure rotation group / [13]. This is the general basis of the relationship of icosahedral symmetry to the solution of the general quintic equation, as Icoaahedr~l symmetry 15 summarized in Klein's book [12]. The icosahedral pure rotation group I is the only non-trivial simple group which can serve as a symmetry point group in three dimensions. This accounts for the unusual appearance of its character table [21]. Consider a group G that is not simple. Then a series of a maximum number of normal subgroups can be assembled, so that C1 = Go ,~ G1 "~"" "~ G,n = G, (2.2) and each factor group Gi/Gi_l is simple. The group G is said to be solvable when each of these factors is of prime order. In this case, each step from Gi to Gi-1 in (2.2) generates radicals of the type ~/~ in the solution of an equation with G as the Galois group, where s is the number of elements in the factor group Gi/Gi-1. Thus equations with solvable Galois groups are exactly those which are solvable using radicals. Conversely, general equations of degree 5 or higher, which have the simple alternating groups An (n > 5) as their Galois groups, are not solvable using only radicals. Some of these and other relevant ideas can be illustrated by the well-known algorithm for solution of the general cubic equation [2] z a-l-a1 x 2 ÷ a2z+a3 = 0. (2.3) The Galois group of Equation (2.3), Sa, is isomorphic to the Dab point group of the equilateral triangle, and has the following normal series and associated factor groups: Go = C1 ~ Ca = Aa '~ Sa = Daa, (2.4) A_s = c3, (2.5a) CI $3 _ C2.